I guess I don’t have any more suggestions other than to do what you can do - and add to it as you have the opportunity. Walking around with you in the arena is more exercise than she presently gets, so that is a plus for now. Since she likes obstacles in hand, see if you can add one simple obstacle as you ride. Build on your successes. Ask for one trot step after a low cavaletti, things like that.
As for “I haven’t done much with her under saddle yet because I know how she’s going to react.” - I know this sounds new-agey, but picture her doing something different. Your body sends cues when you “know” what’s going to happen - so “know” something positive will happen (within reason, of course) and work from there.
Good luck - you have all summer to play with her!